1
Abstract—Smart grid technologies will increase load
management opportunities to customers. As a result, effective
load monitoring is needed to wisely evaluate the load
management options. In response to this need, this paper
presents a solution for microgrid load monitoring using state
estimation techniques based on voltage-sensors installed at the
loads. This solution is special suitable for commercial facilities,
where usually it is difficult to have access to the conductors
responsible for feeding the loads, while the voltages at the
terminal of the loads are easily accessible. A current-based state
estimation algorithm is proposed to estimate the load currents
based on the voltages at the terminals of the loads. Measurements
of currents and power collected from panels are also considered
and provide redundancy for the proposed state estimation
algorithm. A test case is used to show the effectiveness of the
proposed method. The method presents great potential to be
applied in microgrids load monitoring. From the solution
presented in this paper further applications and researches in
this area can be developed.
Index Terms—Microgrids, State estimation, smart grids.
I. INTRODUCTION
With the modernization of the electrical systems, the
concept of microgrids has emerged as one of the solutions for
the future operation of the system as a smart grid. Low-voltage
grids provided with advanced monitoring, control and
automation systems to manage its consumption, generation
and storage can be characterized as microgrids [1]-[2].
Commercial facilities have great potential for the
employment of microgrids solutions. They represent one
segment with significant demand of electricity and potential to
increase efficiency of energy utilization as well as to
participate on demand response programs. Commercial
systems usually present a variety of loads and a high demand
for heating, ventilation, and air conditioning, which usually
can be controlled [3]-[4]. These facilities also deploy small
generators as back-up in case of loss of the utility; additionally
their generation capacity is rising by the increase on solar
photovoltaic generation installed in their systems.
In microgrids, the energy management systems are
responsible for coordinated operation of loads for energy
This work was supported in part by the XX under Grant XX.
X. X. XXX is with XX University, District Hastings, NE 68902 USA (e-
mail: [email protected]).
efficiency improvement and demand response. These
functions can be better performed with detailed power flow
information based on real time measurements. Such
information will allow that a smart algorithm interpret
consumption patterns, which can be used, for instance, to
reduce peak loads, without compromising the customers
comfort, since information about the power consumption of
different segments is known. This information may also allow
monitoring the functioning of fundamental equipment, such as
equipment associated to cooling, ventilation and pumping
functions [4]-[7].
The intuitive solution for monitoring microgrids loads is
the direct measurement of current or power consumed by each
load of the facility installation. However measuring load
current or power in a facility building require access to the
conductors, which can be a difficult task since the power
supply conductors are usually installed in ducts inside the
walls and therefore inaccessible. On the other hand, voltages
at the load terminals are easily accessible and can be measured
by distributed voltage sensors. The voltage sensors should be
able to measure the voltage phasor at each load terminal.
These voltages combined with other measurements available
in the installation can be employed in a state estimation
algorithm to estimate the loads currents. In this context, the
solution for microgrids load monitoring proposed in this paper
is to install voltage sensors at the terminals of the loads and
use these measurements to estimate the currents of the loads.
The treatment of the measurements and the estimation of
the most possible and reasonable values of the system state
can be conducted by a state estimation algorithm. Since
voltage phasors at loads are known, the most appropriate
algorithm for this problem should estimate the currents of
loads, which can be performed by a current-based state
estimation algorithm.
The traditional state estimation methods are applied to
transmission systems and are node-voltage based, that is, the
estimated or state variables are the voltages at the nodes
[8],[9]. The current-based state estimation algorithms present
in the literature have application in power distribution systems
[10]-[13]. In [10] a branch current three-phase state estimation
method was proposed. The algorithm uses the current in the
rectangular form as state variable and it is based on power
injections and power flow measurements. The same state
variables are used for [11] and [12], but they propose a
constant Jacobian matrix by converting the power and
magnitude current measurements to equivalent currents in the
Microgrid Load Monitoring Using State
Estimation Techniques (V1.0)
Author 1, Member, IEEE, and Author 2, Fellow, IEEE
2
rectangular form. In [13] a branch-current based state
estimation is proposed using the magnitude and phase angle of
the branch current as the state variables.
These current-based state estimation algorithms proposed
in the literature cannot be directly applied to estimate the load
currents for this problem. Firstly, the currents estimation
should be based on voltage measurements. Conventional state
estimation algorithms usually employ active and reactive
power measurements to estimate the bus voltages or the
branches currents. Secondly, typically in a facility the main
panel is connected to the secondary side of distribution service
transformer and multiple levels of panels are expanded
downstream until the loads are reached. However, only the
voltages from the load buses and the main panel are known,
since for the most of installations is difficult to have access
and measure the voltage from the panels. Such constrain is a
complicating factor since the currents should be estimated
using only the available voltages.
In this paper a state estimation algorithm is proposed to
estimate the three-phase loads currents of a microgrid. The
algorithm is based on the load voltages and additional
measurements such as power injections and current
magnitudes from the accessible panels. These additional
measurements ensure the state estimation redundancy.
This paper is organized as follows. Section II discusses the
characteristics associated to the typical commercial systems
and the measurements available. Section III describes the
current-based state estimation algorithm proposed. Section IV
presents the test results. Section V summarizes the main
conclusions of this paper.
II. PROBLEM DESCRIPTION
Typically, in a facility the main panel is connected to the
secondary side of distribution service transformer. From the
main panel multiple levels of subpanels are expanded
downstream, and other panels or multiple conductors are
derived from such panels. A typical single-phase equivalent
circuit of a commercial facility is presented in Fig. 1. As can
be seen in this figure, several subpanels are connected to the
main panel. The subpanels are responsible to connect either
other subpanels or the loads. From each one of the bottom
panels a great number of conductors are derived to connect the
loads.
These conductors are confined inside trays or ducts, which
are installed inside the walls. The installation of current or
power measurement devices to monitor the load consumption
requires the access to each one of these conductors. However,
factors such: the difficulty to access the trays or ducts, the
great number of conductors in the same tray, and the difficulty
in identifying or tracking the conductors’ routes became the
installation of power measurement devices impeditive. To
overpass such difficulty, one solution is the measurement of
the voltage at the load terminals. These terminals are easily
accessible and can be measured using plug-in devices. The
voltage measurements of each one of the load terminals can be
synchronized with the voltage from the main panel. So the
voltage magnitude and angle of each load can be considered
known.
Fig. 1. Typical per-phase equivalent facility circuit.
Therefore, once the voltage magnitude and angle at loads
are known, the voltage drop can be used to calculate the load
currents. The configuration of the grid can be easily modeled
by extracting the information from electrical diagrams; and the
electrical impedances from each branch can be calculated
based on the wire gage and length.
The voltages of the intermediate panels or subpanels are
assumed not known, since it may be difficult to access and
measure the voltage from these panels. As the currents
estimation should be based on the voltages at the main panel
and at the loads, a special procedure has to be defined to
compute the load currents with the voltage drop between the
main panel and the loads terminals.
The calculation of the loads current is based on the fact
that all branches currents are a combination of the loads
currents. For example, considering the simple system shown
in Fig. 2, in which the main panel is represented by the bus 1,
the loads are connected to buses 3 and 4. As the voltage at bus
2 is unknown, the equations relating the loads currents to the
voltage drop are:
(1)
(2)
where , and are the phasors of voltage of the main
panel, load 3 and load 4, respectively. and are the load
3 and 4 currents. is the branch current from bus 1 to bus 2.
Z12, Z23 and Z24 are the branches impedances form branch
connecting 1 to 2; 2 to 3 and 2 to 4, respectively.
To solve this system, the number of unknown currents
should be equal to the number of load voltages. So the
solution is to write the branches currents as function of the
load currents. For this example, the branch current is
composed by the sum of the load current with . So, the
branch current can be replaced by a combination of load
Subpanel
Subpanel
Subpanel
Subpanel Subpanel
…
Main panel
Secondary of service
transformer
…
loads
…
loads
…...
…
…...
…...
…...
loads loads loads
loads
Subpanel
Subpanel
Subpanel Subpanel
Subpanel
Subpanel
3
currents, resulting in:
( ) (3)
( ) (4)
Using this assumption, it is possible to calculate all the
currents considering the measured voltages at the main panel
and at the loads terminals. For more complex systems the
solution will require more detailed procedure, which will be
further presented in this paper. This procedure is important for
computing the initial currents estimation for the state
estimation algorithm. The initialization of the variables has a
great impact on the algorithm convergence speed.
Fig. 2. One-line diagram of a simple example system.
Besides the phasor voltages at the loads and at the main
panel, additional measurements are also available from the
main panel and possible from other panels, which can also be
used to estimate the currents. Usually, measurements of active
and reactive power injection are available from the main
panel, the magnitude of some branches currents may also be
available. These measurements should be used with a state
estimation algorithm to estimate the load currents. As already
seen, measurements of voltages in all loads allow to calculate
the load currents, so the extra measurements, such as injected
power and magnitude currents can be used to provide
redundancy to the state estimation algorithm.
III. STATE ESTIMATION FORMULATION
The general form of state estimation problem can be
expressed as:
( ) s.t. c(x)=0 (5)
where z is a m-dimensional vector containing the m
measurements; x is a n-dimensional (n<m) state vector; h(x) is
a m-dimensional vector of functions relating measurements to
state variables; w is a m-dimensional vector containing the
measurement error vector; c(x) is a l-dimensional vector of
functions that model the zero injections as equality
constraints; m is the number of measured quantities; and n is
the number of state variables.
The Weighted Least-Square (WLS) is used to estimate the
state variables, and different weights are selected according to
the accuracy of the measurements, resulting in:
( ) ∑
( ( ))
[ ( )] [ ( )] s.t. c(x) = 0
(6)
where σ2 is the covariance of the measurement and R
-1 is the
inverse matrix of the diagonal matrix with the variances (σ2
i)
in the ith diagonal position, given by:
[
]
(7)
The equality constraints representing the zero injections
measurements are treated by the method of Lagrange
multipliers [14]. The estimated state is obtained by solving the
following system of equations at each iteration:
( ( ) ( )
( ) ) (
)
( ( ) [ ( )]
( ))
(8)
where k is the iteration index, xk is the solution vector at
iteration k, k are the Langrange multipliers at iteration k, H(x)
= (∂h/∂x) and C(x) = (∂c/∂x) are Jacobian matrices and
G(x)=HT(x)R
-1H(x) is the gain matrix.
The problem of State Estimation for microgrids can be
modeled using Fig. 3, which presents a simplified one-phase
facility network. This system is composed by one main panel
and two panels connected downstream from the main. In the
main panel the voltage and the injected power are measured,
the current magnitude may also be measured. In the subpanels
(buses 2 and 3) usually is difficult to have access to measure
the voltage or the power, but sometimes it is possible to
measure the current magnitude. From the last panel before the
load, several circuits are derived to connect the loads. The
terminal voltages are measured, and these measurements are
synchronized with the main panel voltage, making the
magnitude and angle of the loads voltage available. Following
the downstream sequence, numbers are associated to the each
node of the circuit, including panels and load terminals as
nodes, as shown in Fig. 3.
Fig. 3. One-phase diagram Simple system.
A. Measured variables
The loads voltages and the main panel voltage in the
complex domain may be expressed in their rectangular form
by:
1
2
3 4
Main panel
Subpanel
loads
V
0
Z Z
Main Panel 1
V4 V5
Z Z
25
P1+j Q1
I12
I13
1
3 Panel 2 Panel 3
V6
L
V7
I24 I25 I37
I
01
I36
Z
Z
4 5 6 7
I12 I
13
4
(9)
where is the complex voltage and Vq,r and Vq,x are,
respectively, the real and imaginary part of the voltage of a
bus q. The expression of voltage phasor measurements in the
rectangular form improves the convergence properties of the
estimator [15].
Other measurements that also can be collected are the
active and reactive power injection at the main panels, some
branches magnitude currents from the panels. The
measurements vector results in:
[ ] (10)
where Pk and Qk are the active and reactive power injection in
the main panel; Iij is the branch current magnitude from bus i
to j; Vq,r is the real part of the voltage of load connected to bus
q and Vq,x is the imaginary part of the voltage of load
connected to bus q.
B. State variables
The vector x is composed by the currents in the rectangular
form, including the load currents and the branches currents,
such as:
[ ] (11)
where Iij,r and Iij,x are, respectively, the real and imaginary
parts of the current from node i to node j.
C. Functions relating measurement vector to state vector
1) Power injection in the main panel: The equation of the
total power injection in the main panel is given by:
∑
(12)
where Vk is the voltage magnitude of the considered bus, in
this case the main panel and k=1 and Ωk indicates all branches
connected to bus k.
The active power injection is given by:
∑
∑
(13)
The reactive power injection is calculated by:
∑
∑
(14)
2) Equation of the magnitude current branches. The current
magnitude measurement is related to the state variables by:
√( ) ( )
(15)
where Iij is the branch current magnitude, and Iij,r and Iij,x are,
respectively, the real and imaginary parts of the branch current
state variable.
3) Equation of load voltage: The voltage at the load
connected to node q is the voltage at the main panel minus the
voltage drop on the branches between the main panel and the
load. The voltage phasor of the load qth
is given by:
∑
(16)
where is the voltage at the load, is the voltage at the
main panel, zij is the impedance of the branch ij, is the
current of the branch ij and the set Bq contains all branches
where the q-load current flows from the main panels to the
load terminal.
The real part of the voltage is given by:
∑ ( )
(17)
where rij and xij are, respectively, the resistance and reactance
of the branch ij.
The imaginary part is given by:
∑ ( )
(18)
D. Equality Constrains
Zero injection current equality constrains should be
considered for the panels, but the main panel. For the ith
bus,
which is a panel, the real and imaginary parts of the currents
should have a zero injection. For the real part of the current:
∑
(19)
For the imaginary part of the current:
∑
(20)
where Ωi comprises all buses connected to the bus i. The
convention of current signals adopted is currents flowing into
bus i has a positive signal and currents flowing from bus i has
a negative sign.
E. Entries of Jacobin Matrix
The entries of the Jacobian matrix are derived by taking the
differential of the measurement equation with respect to the
state variables.
1) Active power injection measurements: If the branch is
connected to the bus k, for the real part of the current:
(21)
For the imaginary part of the current:
(22)
If the branch is not connected to the bus k, the derivate is
zero.
2) Reactive power injection measurements
If the branch is connected to the bus k, for the real part of
the current:
5
(23)
For the imaginary part of the current:
(24)
If the branch is not connected to the bus k, the derivate is
zero.
3) Current magnitude measurements. For the real part of
the current:
√( ) ( )
(25)
For the imaginary part:
√( ) ( )
(26)
4) Real part of load voltage phasor measurements. If the
current of the load q flows through the branch ij:
For the real part of the current:
(27)
For the imaginary part of the current:
(28)
5) Imaginary part of load voltage phasor measurements:
If the current of the load q flows through the branch ij:
(29)
And for the imaginary part:
(30)
6) Virtual Measurements: Taking the differential of the
equality constrains in relation to the state variables, used to
construct the C matrix, the following equations for the real and
imaginary parts, respectively, are obtained:
∑
( ) (31)
For the imaginary part:
∑
( ) (32)
If the current is flowing into bus i α=2; if the current is
flowing from bus i α=1.
For current-based state estimation algorithms, the inclusion
of the virtual measurements, representing the zero injections
measurements, are discussed in [14]. In this reference the
authors use zero power injection. For an algorithm which the
state variables are the branches currents, it is more efficient
consider zero current injections, which, as can be seen in the
last equations, result in unit elements in the correspondent
Jacobian matrix elements.
F. Initialization
Initialization of the state variables presents a great impact
on the convergence speed of the algorithm. The first value for
the currents can be calculated using the voltages of the loads
terminals and the voltage of the main panel. The voltage drop
between the main panel and the loads terminals can be
calculated using the impedances and currents of each branch
connecting the load to the main panel. However, it is
necessary to consider that only the voltages at the loads are not
enough to compute all the currents, but if each one of the
branches currents is considered a combination of the loads
currents, the loads currents can be firstly calculated and then
all the branches currents can be calculated.
The first step is to identify the branches that connect each
load to the main panel. So, for a load k, Bk contains the set of
all branches connecting load k to the main panel. The second
step is for each one of the branches the load currents that
flows through it should be identified, for example, in Fig. 3 for
the branch from bus 1 to 2, the currents from loads 4 and 5
flows for it. This information can be saved in a set Lij, which
brings the loads that the current flows for the branch ij.
With this information, the load currents can be calculated
by:
[ ] [ ] (33)
where [ ] is the vector containing the difference between the
measured load voltages and the main panel voltage; [ ] is a
vector containing all the loads currents and Z is the impedance
matrix. The matrix Z is given by:
For the elements out of the diagonal:
( ) ( ) ∑
(34)
where zij is the branch impedance which is part of the path of
the load k and of the load q. If Lq and Lk are already known,
Lqk can be calculated by Lq∩Lk. If both loads don’t share any
branch this element will be null.
For the elements of the diagonal:
( ) ∑
(35)
Once the impedance matrix is calculated the load current
can be calculated by:
[ ] ( ) [ ] (36)
One important aspect to highlight is that the impedance
matrix will not change and its inverse can be saved. Once the
load currents are known, a back sweeping procedure can be
applied to calculate all the branches currents, since the
branches currents are a combination of the loads currents.
6
Some state estimation algorithms in the literature that uses
magnitude currents measurements don’t employ these
measurements in the first iteration [10]. These algorithms use
a flat voltage start for first iteration, and in such cases current
magnitude measurements may lead to convergence problems
if used in the first iteration. For current-based state estimation
algorithms a flat voltage start can be used to calculate the first
iteration currents with a backward sweep procedure. In these
cases the current measurements are also excluded in the first
iteration and then introduced in the successive iterations.
Other solution proposed in [11] was reducing the weights of
current magnitude measurements in the first and second
interactions. The initialization procedure proposed in this
paper avoids problems of this nature and additionally speeds
up the algorithm convergence. So, in the algorithm proposed
in this paper all measurements are used with the properly
weights from the first iteration.
G. Algorithm Steps
The algorithm is implemented as the following steps:
Step 1: Define the system configuration and parameters:
Obtain the system configuration, the branches impedances
and the measurements.
Step 2: Initialization: Calculate the first estimation for the
load currents, x(1), based on the measurements of the
voltage at the loads terminals and the main panel voltage.
Step 3: Using back sweeping calculate all branches
currents.
Step 4: Using forward sweeping calculate all nodes
voltages.
Step 5: Calculate the updates of the system state, Δx(n).
Such updates should be calculated based on Equation (8).
Step 6: Update the system state: x(n+1) = x(n)+Δx(n).
Step 7: If Δx(n) is smaller than a convergence tolerance
(stop criterion), then stop. If the number of iteration is
smaller than the maximum iteration number, go to step 4.
If it is not, it does not converge.
In the state estimation formulation proposed, the state
vector is composed of the rectangular form of the currents.
Such choice avoids ill-conditioning problems, especially, for a
state estimation problem whose solution is highly based on
voltage measurements. The voltage measurements are
transformed from the polar form to the rectangular form, with
the same purpose of the use of the current in the rectangular
form. In fact, the use of currents in the rectangular form during
the solution of the state estimation algorithm has been used in
[10]-[12]. One of the differences is that such algorithms are
based on the power and current magnitude measurements,
differently from this presented algorithm.
The use of the currents and voltages in the rectangular form
simplifies the calculation of the Jacobian elements. In fact, for
the voltage measurements, these elements are composed by
the conductors’ reactances and resistances, as can be seen in
Equations (27)-(30).
IV. METHOD VALIDATION
The proposed load currents state estimation method is
implemented for a commercial system. The system is based on
the electrical diagram of a commercial building, which is
presented in Fig. 4. The system comprises, besides the main
panel, 6 panels and 14 three-phase loads. The main panel is
connected to a service transformer, responsible for converting
the voltage from the distribution level to 600 V, which is the
nominal voltage of all loads represented in the system. The
loads are responsible for ventilation, heating and pumping
functions of the building. The data of the system is presented
in the Appendix.
This system is monitored by measuring the phasors of
voltage of all loads as well as the phasor of voltage of the
main panel. The injected active and reactive power of the main
panel and the magnitude of current from the subpanels are also
measured, as indicated in Fig. 4.
Fig. 4. One-line electrical diagram of the test system.
In order to generate the input data for the state estimation
algorithm, a load flow program was employed to obtain, for
the specified loads and main panel voltage, the voltages
magnitude and angle in the loads terminals, the currents and
the injected power of the main panel. Then, the measured
value is calculated by adding a normally distributed
(Gaussian) error on its true value, as following presented:
(37)
where is the measured value,
is the true value
provided by the load flow and ei is the measurement error. The
error is a random number with a standard deviation of the
error (σi), given by rand×(σi). For a given inaccuracy of the
measurement equipment (error%), the standard deviation can
be computed as follows [16]:
(38)
5
loads
6 7
Main panel 1
2
3
4
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
P+jQ
I12 I13
I24 I25 I36 I37
7
For currents and power measurements the considered
error% is 2, and for the voltage measurements the error% is
0.2 [15]. The variance is computed as the square of standard
deviation of the measurements, which values are used to build
the matrix R.
In Table I the average of the absolute errors for the current
magnitude and angles considering 100 simulations are
presented. For each one of these simulations different errors
are associated to the measurements, since the errors are
created by a random function. The convergence criteria
applied to the state estimation algorithm considers that the
maximum update of the state variable should be smaller than
1.10-4
pu. As the currents resulted from the algorithm are in
the rectangular form, they were converted to the polar form
and compared with the original values obtained by the power
flow program.
TABLE I
AVERAGE OF THE ABSOLUTE ERROR OF THE CURRENT AND ANGLE
Branch or
load
Mag.
(pu)
Angle
(deg.)
1 (1-2) 0.0086 0.2403
2 (1-3) 0.0100 0.3632
3 (2-4) 0.0077 0.5610
4 (2-5) 0.0053 0.4175
5 (3-6) 0.0051 0.5920
6 (3-7) 0.0069 0.4627
8 0.0118 0.8481
9 0.0110 0.7337
10 0.0115 1.5075
11 0.0117 1.9886
12 0.0113 0.7588
13 0.0115 0.7156
14 0.0129 0.8281
15 0.0131 1.5033
16 0.0105 1.4781
17 0.0113 1.5158
18 0.0111 1.7082
19 0.0111 0.5782
20 0.0122 1.0841
21 0.0121 0.7524
22 0.0104 0.7838
For the 100 simulated cases, the initialization of the
program improved the convergence process and all simulated
cases converged. The most cases converged with 3 iterations,
and the maximum number of iterations was 4.
As can be seen from TABLE I, the currents from the panels
present lower average errors than the loads currents. This
behavior is explained by the magnitude current measurements
installed in these panels. Although current magnitude
measurements from the panels were considered to obtain the
results, in case of these measurements are not available, it
won’t interfere in the observability of the system. The
observability is ensured by the voltages measurements and by
the virtual measurements, that is, the zero injection current
equality constrains of the panels. If the current flow
magnitudes are not considered the only redundancy
measurement will be the active and reactive power measured
at the main panel.
The results for a second case (Case II) are presented in
Table II, which shows the average of the absolute errors of
current magnitudes and angles for 100 simulations. In these
simulations no current magnitude measurements in the panels
were considered. As one can notice, comparing Table II to
Table I, the average errors for the loads (8-22) are very
similar, while the panels branches currents (1-6) presented a
higher error in Case II. TABLE II
AVERAGE OF THE ABSOLUTE ERROR OF THE CURRENT AND ANGLE – CASE
II
Branch or
load
Mag.
(pu)
Angle
(deg.)
1 (1-2) 0.0096 0.2431
2 (1-3) 0.0153 0.5172
3 (2-4) 0.0106 0.6931
4 (2-5) 0.0116 0.3623
5 (3-6) 0.0101 0.6100
6 (3-7) 0.0146 0.7033
8 0.0111 0.9409
9 0.0104 0.6357
10 0.0112 1.4913
11 0.0112 1.8498
12 0.0128 0.7977
13 0.0137 0.8013
14 0.0132 0.7868
15 0.0135 1.4796
16 0.0134 1.8120
17 0.0105 1.3460
18 0.0120 1.8296
19 0.0115 0.7278
20 0.0106 1.1198
21 0.0130 0.9770
22 0.0116 0.9489
Fig. 5 presents the comparison of the panels current
magnitude relative errors between Case I and Case II. For
Case I, the panels current magnitude measurements are
considered in the state estimation algorithm and for Case II
these measurements are not considered. As one can notice,
taking the panels current magnitude measurements into
consideration slightly improve the estimation of the currents
of the panels. So, if current magnitude measurements from the
panels are available it is worth to include such measurements
in the state estimation.
Fig. 5. Comparison of relative errors of the estimated current magnitude.
1 2 3 4 5 60
1
2
3
4
branch number
erro
r per
centa
ge
(%)
Case I
Case II
8
V. CONCLUSIONS
This paper has presented a new method for microgrid load
monitoring. The main idea of the proposed method is to
measure the phasors voltages at the terminals of the loads,
which comprises a good solution for facilities which the load
conductors are inaccessible. A current-based state estimation
algorithm is proposed to estimate the currents based on
voltage measurements and other measurements available in
the facility. The method was applied to a test system based on
a real system and the results revealed that the method can be
applied to microgrids load monitoring. From the solution
presented in this paper further applications and researches in
this area can be developed and proposed. The proposed
voltage-sensor based load monitoring can also be extended
and adapted to be applied in residential installations.
VI. APPENDIX
The data of the system presented in Fig. 4 is presented at
Table III and the powers of the loads are presented at Table
IV. A base power of 1MVA is used and a base voltage of 600
V. TABLE III
CONDUCTORS PARAMETERS
From To R(pu) X(pu)
1 2 0.0007 0.0058
1 3 0.0007 0.0056
2 4 0.0031 0.0047
2 5 0.0043 0.0067
3 6 0.0059 0.0092
3 7 0.0041 0.0063
4 8 0.0334 0.0188
4 9 0.0352 0.0198
4 10 0.0220 0.0253
5 11 0.0224 0.0257
5 12 0.0349 0.0197
5 13 0.0342 0.0193
5 14 0.0302 0.0170
5 15 0.0213 0.0245
6 16 0.0202 0.0232
6 17 0.0208 0.0239
6 18 0.0231 0.0266
7 19 0.0356 0.0201
7 20 0.0355 0.0200
7 21 0.0346 0.0195
7 22 0.0337 0.0190
TABLE IV
LOAD PARAMETERS
Bus P(kW) Q(kvar)
8 139.2 53.6
9 149.2 63.6
10 198.9 79.7
11 168.9 59.7
12 128.2 53.0
13 120.9 53.2
14 138.0 58.5
15 178.9 80.8
16 230.0 80.8
17 204.0 80.8
18 197.2 69.7
19 178.9 69.7
20 129.2 43.2
21 139.2 53.2
22 150.2 53.6
VII. REFERENCES
[1] F. Katiraei, R. Iravani, N. Hatziargyriou, and A. Dimeas, “Microgrids
Management,” IEEE Power & Energy Magazine, vol. 6 pp. 54-65.
May/June 2008
[2] H. S. V. S. K. Nunna and S. Doolla, “Energy Management in Microgrids
Using Demand Response and Distributed Storage—A Multiagent
Approach,” IEEE Trans. Power Delivery, vol. 28, pp. 939-947. Apr.
2013.
[3] “Buildings Energy Data Book, Chapter 3: Commercial Sector,” U.S.
Department of Energy [Online]. Available:
http://buildingsdatabook.eren.doe.gov/default.aspx
[4] P. Palensky and D. Dietrich, “Demand Side Management: Demand
Response, Intelligent Energy Systems, and Smart Loads,” IEEE Trans.
Industrial Informatics, vol. 7, pp. 381-388. Aug. 2011.
[5] D. Dietrich, D. Bruckner, G. Zucker, and P. Palensky, “Communication
and Computation in Buildings: A Short Introduction and Overview,”
IEEE Trans. Industrial Electronics, vol. 57, pp. 3577-3584. Nov. 2010.
[6] Y. Du, L. Du, B. Lu, R. Harley, and T. Habetler, “A Review of
Identification and Monitoring Methods for Electric Loads in
Commercial and Residential Buildings,” presented at the Energy
Conversion Congress and Exposition, Atlanta, USA, 2010.
[7] J. L. Mathieu, P. N. Price, S. Kiliccote, and M. A. Piette, “Quantifying
Changes in Building Electricity Use With Application to Demand
Response,” IEEE Trans. Smart Grid, vol. 2, pp. 507-518. Sep. 2011.
[8] A. Monticelli, State Estimation in Electric Power System: A Generalized
Approach, Kluwer Academic Publishers: USA, 1999.
[9] A. Abur and A. G. Exposito, Power System State Estimation: Theory
and Implementation, Marcel-Dekker: New York, USA, 2004.
[10] M. E. Baran and A.W. Kelley, “A branch-current-based state estimation
method for distribution systems,” IEEE Trans. Power Syst., vol. 10, pp.
483–491, Feb. 1995.
[11] C. N. Lu, J. H. Teng, and W. H. E. Liu, “Distribution system state
estimation,” IEEE Trans. Power Syst., vol. 10, pp. 229–240, Feb. 1995.
[12] W. M. Lin, J. H. Teng, and S. J. Chen, “A highly efficient algorithm in
treating current measurements for the branch-current-based distribution
state estimation,” IEEE Trans. Power Delivery, vol. 16, pp. 433–439,
Jul. 2001.
[13] H. Wang and N. N. Schulz, “A Revised Branch Current-Based
Distribution System State Estimation Algorithm and Meter Placement
Impact,” IEEE Trans. Power Systems, vol. 19, pp. 207-213, Feb. 2004.
[14] W. M. Lin and J. H. Teng “State estimation for distribution systems with
zero-injection constraints,” IEEE Trans. Power Syst., vol. 11, pp. 518–
524, Feb. 1996.
[15] G. N. Korres and N. M. Manousakis, “State Estimation and
Observability Analysis for Phasor Measurement Unit Measured
Systems,” IET Gener., Transm. & Distrib., vol. 6, pp. 902-913. Sep.
2012.
[16] R. Singh, B. C. Pal, and R. A. Jabr, “Choice of Estimator for
Distribution System State Estimation,” IET Gener., Transm. & Distrib.,
vol. 3, pp. 666-678. Jul. 2009.
VIII. BIOGRAPHIES
XXXX (M’1888, F’17) xxxx
1
Abstract — A major feature of the smart grid is its diverse
load management opportunities for customers, which calls for
innovative techniques for load monitoring. This paper presents a
method to monitor microgrid loads using a set of voltage sensors
and state estimation algorithms. The technique is especially
suited for commercial facility based microgrids where it is often
difficult to measure loads directly but the load voltages can be
sensed. A current-based state estimation algorithm is proposed to
estimate the load currents using the voltages sensed at the
terminals of the loads. Currents and powers collected from
limited number of panels are also utilized to provide redundancy
for estimation. Case studies have shown that the proposed
method represents a promising alternative direction for
monitoring microgrid loads.
Index Terms—Microgrids, load monitoring, state estimation,
smart grids.
I. INTRODUCTION
ITH the modernization of the electrical systems, the
concept of microgrids has emerged as one of the
solutions for the future operation of the system as a smart grid.
Low-voltage grids provided with advanced monitoring,
control and automation systems to manage its consumption,
generation and storage can be characterized as microgrids [1]-
[2].
One of the representative microgrids is the commercial
facilities, as they are owned by single owners and thus are
much easier to setup microgrid operation. They also represent
one energy-use segment with significant potential to increase
energy efficiency and to participate in demand response.
Commercial systems usually have a wide variety of loads.
They can also deploy small generators using renewable
sources [3]-[4]. For microgrid operations, the energy
This work was supported in part by CNPq and FAPESP of Brazil and
NSERC of Canada.
A. P. Grilo is with University of Alberta, Edmonton, AB T6G 2V4,
Canada, as Post-Doctoral Fellow, and Federal University of ABC, Santo
Andre, SP, Brazil, as Assistant Professor on a leave of absence (e-mail:
W. Xu is with the Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:
M. C. de Almeida is with the Department of Electrical Energy Systems,
University of Campinas, Campinas, 13083-852 Brazil (e-mail:
management systems are responsible for coordinated
operation of loads and generators. A critical piece of
information needed by the system is the real-time power
consumption of various loads. In addition, this information
can be used for equipment condition monitoring, which is
another smart feature of the microgrids [4]-[7].
The direct solution to monitoring loads is to measure
current or power consumed by each load of the facility.
However measuring load current or power requires accessing
to the conductors supplying the loads. This can be difficult or
even impossible for facilities that have already been built. This
is because the power supply conductors are usually installed in
ducts inside the walls and therefore inaccessible. As a result,
except for a few large loads with built-in sensors, most of the
loads in a commercial building cannot be monitored at
present.
On the other hand, voltages at the load terminals are often
accessible and can be measured by distributed voltage sensors.
These voltages, combined with the network topologies and
parameters, can be used to estimate the load currents through
state-estimation-like algorithms. Based on this reasoning, a
distributed voltage sensor based, computational load
monitoring solution is proposed in this paper. By utilizing the
work done in the area of power system state estimation, this
paper has shown that the proposed technique represents a
promising solution to the microgrid load monitoring problem.
This paper is organized as follows. Section II discusses the
challenges of load monitoring for commercial microgrids and
defines the problem to be solved. It also reviews the various
state estimation algorithms that may be adopted for the load
monitoring problem. Section III presents the proposed current-
based state estimation algorithm. Section IV shows the case
study results. Section V summarizes the main conclusions of
this paper.
II. PROBLEM DESCRIPTION
The common electrical configuration of a commercial
facility starts at the main panel. The voltage, current and
power are often monitored at this location. From the main
panel multiple levels of subpanels are expanded downstream,
and other panels or multiple conductors are derived from such
panels. A typical equivalent circuit of a commercial facility is
shown in Fig. 1. As can be seen in this figure, several
subpanels are connected to the main panel. The subpanels are
responsible to connect either other subpanels or the loads.
Microgrid Load Monitoring Using State
Estimation Techniques (Final Version)
Ahda P. Grilo, Member, IEEE, Wilsun Xu, Fellow, IEEE,
and Madson C. de Almeida, Member, IEEE
W
2
From each one of the bottom panels a great number of
conductors are used to supply various loads.
These conductors are confined inside trays or ducts, which
are installed inside the walls. The installation of current
measurement devices to monitor the load consumption
requires the access to each of these conductors. However,
factors such as the difficulty to access the trays or ducts, the
great number of conductors in the same tray, and the difficulty
in identifying or tracing the conductors’ routes make the
installation of power measurement devices very difficult and
costly. On the other hand, the load voltages can be measured
relatively easily since many receptacles are available in a
facility. One can, for example, develop low cost voltage
sensors and install them at various locations. These sensors are
synchronized and networked. It is possible to estimate the load
currents from such measured voltage phasors if the network
topology and conductor impedances are known.
Fig. 1. Typical per-phase equivalent facility circuit.
The topology and impedance data can be obtained from the
building electrical diagrams. The calculation of the loads
current is based on the fact that all branches currents are a
combination of the loads currents. For example, considering
the simple system shown in Fig. 2, in which the main panel is
represented by the bus 1, the loads are connected to buses 3
and 4. As the voltage at bus 2 is unknown, the equations
relating the loads currents to the voltage drop are:
(1)
(2)
where , and are the phasors of voltage of the main
panel, load 3 and load 4, respectively. and are the load
3 and 4 currents. is the branch current from bus 1 to bus 2.
Z12, Z23 and Z24 are the branches impedances from branch
connecting 1 to 2; 2 to 3 and 2 to 4, respectively.
To solve this system, the number of unknown currents
should be equal to the number of load voltages. So the
solution is to write the branches currents as function of the
load currents. For this example, the branch current is
composed by the sum of the load current with . So, the
branch current can be replaced by a combination of load
currents, resulting in:
( ) (3)
( ) (4)
Using this transform, it is possible to calculate all the
currents considering the measured voltages at the main panel
and at the loads terminals. For more complex systems the
solution will require more detailed procedure, which will be
further presented in this paper.
Fig. 2. One-line diagram of a simple example system.
Besides the phasor voltages at the loads and at the main
panel, additional measurements are also available from the
main panel and possibly from other panels, which can also be
used to estimate the currents. Usually, measurements of active
and reactive power injection are available from the main
panel, the magnitude of some branches currents may also be
available. These measurements can be used to increase
redundancy for the estimation algorithm. The voltages of the
intermediate panels or subpanels are assumed not known,
since they are difficult to access for measurement. As the
currents estimation should be based on the voltages at the
main panel and at the loads, a special procedure has to be
defined to compute the load currents with the voltage drop
between the main panel and the loads terminals.
The problem to be solved can, therefore, be defined as an
estimation problem: solve for load currents based on multiple
voltage and power measurements at various locations. This
problem is closely related to the well-known power system
state estimation problem [8],[9]. The difference here is that the
load monitoring problem is to estimate currents (or power)
from voltages whereas the traditional state estimation problem
estimates voltages from powers (or currents).
The majority traditional state estimation methods published
focus on the estimation of transmission system voltages.
Nodal voltages are the unknown to estimate [8],[9]. Reference
[10]-[13] presented current-based state estimation algorithms
for power distribution systems. For example, reference [10]
uses the currents in rectangular coordinates as state variables
and power injections as observable quantities. The same state
variables are used for [11] and [12], but they propose a
constant Jacobian matrix solution algorithm. In [13] a branch-
current based state estimation is proposed using the magnitude
and phase angle of the branch current as the state variables.
Subpanel
Subpanel
Subpanel
Subpanel Subpanel
…
Main panel
Secondary of service
transformer
…
loads
…
loads
…...
…
…...
…...
…...
loads loads loads
loads
Subpanel
Subpanel
Subpanel Subpanel
Subpanel
Subpanel
1
2
3 4
Main panel
Subpanel
loads
3
The above current-based state estimation algorithms are
more closely related to the load monitoring problem. But they
cannot be directly applied. Firstly, the observable quantities
are the voltage measurements here. Conventional state
estimation algorithms usually employ active and reactive
power measurements to estimate the bus voltages or the
branches currents. Secondly, a facility’s main panel is
connected to the secondary side of distribution service
transformer and multiple levels of panels are expanded
downstream until the loads are reached. Only the voltages
from the load buses and the main panel are known. Such
constraint is a complicating factor since the currents should be
estimated using only the available voltages. To solve the
problem of load monitoring, a new state estimation problem is
formulated and new algorithms are proposed to solve it.
III. STATE ESTIMATION FORMULATION
The general form of state estimation problem can be
expressed as:
( ) s.t. c(x)=0 (5)
where z is a m-dimensional vector containing the m
measurements; x is a n-dimensional (n<m) state vector; h(x) is
a m-dimensional vector of functions relating measurements to
state variables; w is a m-dimensional vector containing the
measurement error vector; c(x) is a l-dimensional vector of
functions that model the zero injections as equality
constraints; m is the number of measured quantities; and n is
the number of state variables.
The Weighted Least-Square (WLS) is used to estimate the
state variables, and different weights are selected according to
the accuracy of the measurements, resulting in:
( ) ∑
( ( ))
[ ( )] [ ( )] s.t. c(x) = 0
(6)
where σ2 is the covariance of the measurement and R
-1 is the
inverse matrix of the diagonal matrix with the variances (σ2i)
in the ith
diagonal position, given by:
[
]
(7)
The equality constraints representing the zero injections
measurements are treated by the method of Lagrange
multipliers [14]. The estimated state is obtained by solving the
following system of equations at each iteration:
( ( ) ( )
( ) ) (
)
( ( ) [ ( )]
( ))
(8)
where k is the iteration index, xk is the solution vector at
iteration k, k are the Langrange multipliers at iteration k, H(x)
= (∂h/∂x) and C(x) = (∂c/∂x) are Jacobian matrices and
G(x)=HT(x)R
-1H(x) is the gain matrix.
The equations of State Estimation for microgrids can be
formulated using Fig. 3 as example, which presents a
simplified one-phase facility network. This system is
composed by one main panel and two panels connected
downstream from the main. In the main panel the voltage and
the injected power are measured, the current magnitude may
also be measured. In the subpanels (buses 2 and 3) usually is
difficult to have access to measure the voltage or the power,
but sometimes it is possible to measure the current magnitude.
From the bottom panels, several circuits are derived to connect
the loads. The terminal voltages are measured, and these
measurements are synchronized with the main panel voltage,
making the magnitude and angle of the loads voltage
available. Following the downstream sequence, numbers are
associated to the each node of the circuit, including panels and
load terminals as nodes, as shown in Fig. 3.
Fig. 3. One-phase diagram Simple system.
A. Measured variables
The loads voltages and the main panel voltage in the
complex domain may be expressed in their rectangular form
by:
(9)
where is the complex voltage and Vq,r and Vq,x are,
respectively, the real and imaginary part of the voltage of a
bus q. The expression of voltage phasor measurements in the
rectangular form improves the convergence properties of the
estimator [15].
Other measurements that also can be collected are the
active and reactive power injection at the main panel and some
branches magnitude currents from the panels. The
measurements vector results in:
[ ] (10)
where Pk and Qk are the active and reactive power injection in
the main panel; Iij is the branch current magnitude from bus i
to j; Vq,r is the real part of the voltage of load connected to bus
q and Vq,x is the imaginary part of the voltage of load
connected to bus q.
B. State variables
The vector x is composed by the currents in the rectangular
V
0
Z Z
Main Panel 1
V4 V5
Z Z
25
P1+j Q1
I12
I13
1
3 Panel 2 Panel 3
V6
L
V7
I24 I25 I37
I
01
I36
Z
Z
4 5 6 7
I12 I
13
4
form, including the load currents and the branches currents,
such as:
[ ] (11)
where Iij,r and Iij,x are, respectively, the real and imaginary
parts of the current from node i to node j.
C. Functions relating measurement vector to state vector
1) Power injection in the main panel: The equation of the
total power injection in the main panel is given by:
∑
(12)
where Vk is the voltage magnitude of the considered bus, if the
power injection is at the main panel k=1, and Ωk indicates the
set of all branches connected to bus k. So, the active power
injection is given by:
∑
∑
(13)
The reactive power injection is calculated by:
∑
∑
(14)
2) Equation of the magnitude current branches. The current
magnitude measurement is related to the state variables by:
√( ) ( )
(15)
where Iij is the measured branch current magnitude, and Iij,r
and Iij,x are, respectively, the real and imaginary parts of the
branch current state variable.
3) Equation of load voltage: The voltage at the load
connected to node q is the voltage at the main panel minus the
voltage drop on the branches between the main panel and the
load. So, it is necessary the information about the branches
that connect each load to the main panel, which for the load q
is defined as the set Bq. Such set contains all branches that are
path of the current of load q from the main panel to the load.
For example, in Fig. 3, the set B4 for load 4 contains the
branches 1-2 and 2-4.
The voltage phasor of the load qth
is given by:
∑
(16)
where is the voltage at the load, is the voltage at the
main panel, zij is the impedance of the branch ij, is the
current of the branch ij and the set Bq contains all branches
connecting load q to the main panel.
The real part of the voltage is given by:
∑( )
(17)
where rij and xij are, respectively, the resistance and reactance
of the branch ij.
The imaginary part is given by:
∑( )
(18)
D. Equality Constrains
Zero injection current equality constrains should be
considered for the panels, but the main panel. For the ith
bus,
which is a panel, the real and imaginary parts of the currents
should have a zero injection. For the real part of the current:
∑
(19)
For the imaginary part of the current:
∑
(20)
where Ωi comprises all buses connected to the bus i. The
adopted convention of current signals is currents flowing into
bus i has a positive signal and currents flowing from bus i has
a negative sign.
E. Entries of Jacobin Matrix
The entries of the Jacobian matrix are derived by taking the
differential of the measurement equation with respect to the
state variables.
1) Active power injection measurements: If the branch is
connected to the bus k, for the real part of the current:
(21)
For the imaginary part of the current:
(22)
If the branch is not connected to the bus k, the derivate is
zero.
2) Reactive power injection measurements
If the branch is connected to the bus k, the differential with
respect to the the real part of the current:
(23)
For the imaginary part of the current:
(24)
If the branch is not connected to the bus k, the derivate is
zero.
3) Current magnitude measurements. For the real part of
the current:
√( ) ( )
(25)
For the imaginary part:
5
√( ) ( )
(26)
4) Real part of load voltage phasor measurements. If the
branch ij is part of the path of the current of the load q, or if
the branch ij belongs to the set Bq, the differential with respect
to the real part of the current is:
(27)
For the imaginary part of the current it results in:
(28)
5) Imaginary part of load voltage phasor measurements:
If the branch ij is part of the path of the current of the load
q, the differential with respect to the real part of the current is:
(29)
And for the imaginary part:
(30)
6) Virtual Measurements: Taking the differential of the
equality constrains in relation to the state variables, used to
construct the C matrix, the following equations for the real and
imaginary parts, respectively, are obtained:
∑
( ) (31)
For the imaginary part:
∑
( ) (32)
If the current is flowing into bus i α=2; if the current is
flowing from bus i α=1.
For current-based state estimation algorithms, the inclusion
of the virtual measurements, representing the zero injections
measurements, are discussed in [14]. In this reference the
authors use zero power injection. For an algorithm which the
state variables are the branches currents, it is more efficient
consider zero current injections, which, as can be seen in the
last equations, result in unit elements in the correspondent
Jacobian matrix elements.
F. Initialization
Initialization of the state variables presents a great impact
on the convergence speed of the algorithm. The first value for
the currents can be calculated using the voltages of the loads
terminals and the voltage of the main panel. The voltage drop
between the main panel and the loads terminals can be
calculated using the impedances and currents of each branch
connecting the load to the main panel. However, it is
necessary to consider that only the voltages at the loads are not
enough to compute all the currents, the currents from
branches, which no loads are connected, can be considered a
combination of the currents of loads. So, the currents of the
loads can be firstly calculated and then all the currents of the
other branches can be calculated using back sweeping
procedure.
The currents of the loads can be calculated by:
[ ] [ ] (33)
where [ ] is the vector containing the difference between the
main panel voltage and the measured load voltages, [ ] is a
vector containing all the loads currents and Z is the impedance
matrix.
The calculation of the elements of the impedance matrix,
which relates the currents of the loads to the voltage drop from
the main panel to the loads, depends on information about the
branches that connect each load to the main panel. The set Bq,
already defined, contains the branches connecting load q to the
main panel. The nomenclature used so far is the load q is the
one connected to the bus q. However, for the impedance
matrix, the numbers associated to the loads should be in
accordance with the sequence of the voltage and currents of
the loads in the vectors [ ] and [ ]. So the load p
corresponds to the pth
load in the current or voltage drop
vector. So, the elements of the main diagonal of the matrix Z
are given by:
( ) ∑
(34)
where Bp is a set including the branches that connect the pth
load to the main panel.
The elements out of the main diagonal are given by:
( ) ( ) ∑
(35)
where Bpt is a set including the branches shared by currents
from loads p and t. If Bp and Bt are already known, Bpt can be
calculated by Bp∩Bt. If both loads don’t share any branch this
element will be null.
Once the impedance matrix is calculated the load current
can be calculated by:
[ ] ( ) [ ] (36)
One important aspect to highlight is that the impedance
matrix is constant if the system topology is maintained.
Therefore the impedance matrix can be inverted or factored
offline, reducing the computational effort.
Some state estimation algorithms in the literature, which
use magnitude currents measurements, don’t employ these
measurements in the first iteration [10]. These algorithms use
a flat voltage start for first iteration, and in such cases current
magnitude measurements may lead to convergence problems
if used in the first iteration. For current-based state estimation
algorithms a flat voltage start can be used to calculate the first
6
iteration currents with a backward sweep procedure. In these
cases the current measurements are also excluded in the first
iteration and then introduced in the successive iterations.
Other solution proposed in [11] was reducing the weights of
current magnitude measurements in the first and second
interactions. The initialization procedure proposed in this
paper avoids problems of this nature and additionally speeds
up the algorithm convergence. Besides, all measurements are
used with the properly weights from the first iteration.
G. Algorithm Steps
The algorithm is implemented as the following steps:
Step 1: Define the system configuration and parameters:
Obtain the system configuration, the branches impedances
and the measurements.
Step 2: Initialization: Calculate the first estimation for the
load currents, x(1), based on the measurements of the
voltage at the loads terminals and the main panel voltage.
Step 3: Using back sweeping calculate all branches
currents.
Step 4: Using forward sweeping calculate all nodes
voltages.
Step 5: Calculate the updates of the system state, Δx(n).
Such updates should be calculated based on Equation (8).
Step 6: Update the system state: x(n+1) = x(n)+Δx(n).
Step 7: If Δx(n) is smaller than a convergence tolerance
(stop criterion), then stop. If the number of iteration is
smaller than the maximum iteration number, go to step 4.
If it is not, it does not converge.
In the state estimation formulation proposed, the state
vector is composed of the rectangular form of the currents.
Such choice avoids ill-conditioning problems, especially, for a
state estimation problem whose solution is highly based on
voltage measurements. The voltage measurements are
transformed from the polar form to the rectangular form, with
the same purpose of the use of the current in the rectangular
form. Actually, the use of currents in the rectangular form
during the solution of the state estimation algorithm has been
used in [10]-[12]. One of the differences is that such
algorithms are based on the power and current magnitude
measurements, differently from the presented algorithm.
The use of the currents and voltages in the rectangular form
simplifies the calculation of the Jacobian elements. In fact, for
the voltage measurements, these elements are composed by
the conductors’ reactances and resistances, as can be seen in
Equations (27)-(30).
IV. METHOD VALIDATION
The proposed load currents state estimation method is
implemented for a commercial system. The system is based on
the electrical diagram of a commercial building, which is
presented in Fig. 4. The system comprises, besides the main
panel, 6 panels and 14 three-phase loads. The main panel is
connected to a service transformer, responsible for converting
the voltage from the distribution level to 600 V, which is the
nominal voltage of all loads represented in the system. The
loads are responsible for ventilation, heating and pumping
functions of the building. The data of the system is presented
in the Appendix.
This system is monitored by measuring the phasors of
voltage of all loads as well as the phasor of voltage of the
main panel. The injected active and reactive power of the main
panel and the magnitude of current from the subpanels are also
measured, as indicated in Fig. 4.
In order to generate the input data for the state estimation
algorithm, a load flow program was employed to obtain, for
the specified loads and main panel voltage, the voltages
magnitude and angle in the loads terminals, the currents and
the injected power of the main panel. Then, the measured
value is calculated by adding a normally distributed
(Gaussian) error on its true value, as following presented:
(37)
where is the measured value,
is the true value
provided by the load flow and ei is the measurement error. The
error is a random number with a standard deviation of the
error (σi), given by rand×(σi). For a given inaccuracy of the
measurement equipment (error%), the standard deviation can
be computed as follows [16]:
(38)
For currents and power measurements the considered
error% is 2, and for the voltage measurements the error% is
0.2 [15]. The variance is computed as the square of standard
deviation of the measurements, which values are used to build
the matrix R.
Fig. 4. One-line electrical diagram of the test system.
In Table I the average of the absolute errors for the current
magnitude and angles considering 100 simulations are
presented. For each one of these simulations different errors
are associated to the measurements, since the errors are
created by a random function. The convergence criteria
applied to the state estimation algorithm considers that the
maximum update of the state variable should be smaller than
P+jQ
5
loads
6 7
Main panel 1
2
3
4
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
I12 I13
I24 I25 I36 I37
7
1.10-4
pu. As the currents resulted from the algorithm are in
the rectangular form, they were converted to the polar form
and compared with the original values obtained by the power
flow program.
For the 100 simulated cases, the initialization of the
program improved the convergence process and all simulated
cases converged. The most cases converged with 3 iterations,
and the maximum number of iterations was 4.
As can be seen from Table I, the currents from the panels
present lower average errors than the loads currents. This
behavior is explained by the magnitude current measurements
installed in these panels. Although current magnitude
measurements from the panels were considered to obtain the
results, in case of these measurements are not available, it
won’t interfere in the observability of the system. The
observability is ensured by the voltages measurements and by
the virtual measurements, that is, the zero injection current
equality constrains of the panels. If the current flow
magnitudes are not considered the only redundancy
measurement will be the active and reactive power measured
at the main panel. TABLE I
AVERAGE OF THE ABSOLUTE ERROR OF THE CURRENT AND ANGLE
Branch or
load
Mag.
(pu)
Angle
(deg.)
1 (1-2) 0.0086 0.2403
2 (1-3) 0.0100 0.3632
3 (2-4) 0.0077 0.5610
4 (2-5) 0.0053 0.4175
5 (3-6) 0.0051 0.5920
6 (3-7) 0.0069 0.4627
8 0.0118 0.8481
9 0.0110 0.7337
10 0.0115 1.5075
11 0.0117 1.9886
12 0.0113 0.7588
13 0.0115 0.7156
14 0.0129 0.8281
15 0.0131 1.5033
16 0.0105 1.4781
17 0.0113 1.5158
18 0.0111 1.7082
19 0.0111 0.5782
20 0.0122 1.0841
21 0.0121 0.7524
22 0.0104 0.7838
The results for a second case (Case II) are presented in
Table II, which shows the average of the absolute errors of
current magnitudes and angles for 100 simulations. In these
simulations no current magnitude measurements in the panels
were considered. As one can notice, comparing Table II to
Table I, the average errors for the loads (8-22) are very
similar, while the panels branches currents (1-6) presented a
higher error in Case II.
TABLE II
AVERAGE OF THE ABSOLUTE ERROR OF CURRENT AND ANGLE – CASE II
Branch or
load
Mag.
(pu)
Angle
(deg.)
1 (1-2) 0.0096 0.2431
2 (1-3) 0.0153 0.5172
3 (2-4) 0.0106 0.6931
4 (2-5) 0.0116 0.3623
5 (3-6) 0.0101 0.6100
6 (3-7) 0.0146 0.7033
8 0.0111 0.9409
9 0.0104 0.6357
10 0.0112 1.4913
11 0.0112 1.8498
12 0.0128 0.7977
13 0.0137 0.8013
14 0.0132 0.7868
15 0.0135 1.4796
16 0.0134 1.8120
17 0.0105 1.3460
18 0.0120 1.8296
19 0.0115 0.7278
20 0.0106 1.1198
21 0.0130 0.9770
22 0.0116 0.9489
Fig. 5 presents the comparison of the panels current
magnitude relative errors between Case I and Case II. For
Case I, the panels current magnitude measurements are
considered in the state estimation algorithm and for Case II
these measurements are not considered. As one can notice,
taking the panels current magnitude measurements into
consideration slightly improve the estimation of the currents
of the panels. So, if current magnitude measurements from the
panels are available it is worth to include such measurements
in the state estimation.
Fig. 5. Comparison of relative errors of the estimated current magnitude.
V. CONCLUSIONS
This paper has presented a novel and attractive approach
for microgrid load monitoring. The main idea of the proposed
method is to use easily accessible voltage measurements to
estimate load currents. The estimation method is based on the
state estimation algorithms. The proposed technique represents
a good solution for microgrid facilities where the load
conductors are inaccessible for current sensing. The proposed
current-based state estimation algorithm has been applied to a
test system and the results revealed that it is a promising
alternative direction for monitoring microgrid loads. The idea
1 2 3 4 5 60
1
2
3
4
branch number
err
or
perc
enta
ge (
%)
Case I
Case II
8
of using voltages to estimate currents as presented in this
paper has some other applications. For example, it could be
used to monitor home appliance behavior by using distributed
voltage sensors installed at various locations of a home.
VI. APPENDIX
The data of the system presented in Fig. 4 is presented at
Table III and the powers of the loads are presented at Table
IV. The nominal power is 1MVA and the nominal voltage is
600 V. TABLE III
CONDUCTORS PARAMETERS
From To R (pu) X (pu)
1 2 0.0007 0.0058
1 3 0.0007 0.0056
2 4 0.0031 0.0047
2 5 0.0043 0.0067
3 6 0.0059 0.0092
3 7 0.0041 0.0063
4 8 0.0334 0.0188
4 9 0.0352 0.0198
4 10 0.0220 0.0253
5 11 0.0224 0.0257
5 12 0.0349 0.0197
5 13 0.0342 0.0193
5 14 0.0302 0.0170
5 15 0.0213 0.0245
6 16 0.0202 0.0232
6 17 0.0208 0.0239
6 18 0.0231 0.0266
7 19 0.0356 0.0201
7 20 0.0355 0.0200
7 21 0.0346 0.0195
7 22 0.0337 0.0190
TABLE IV
LOAD PARAMETERS
Bus P(kW) Q(kvar)
8 139.2 53.6
9 149.2 63.6
10 198.9 79.7
11 168.9 59.7
12 128.2 53.0
13 120.9 53.2
14 138.0 58.5
15 178.9 80.8
16 230.0 80.8
17 204.0 80.8
18 197.2 69.7
19 178.9 69.7
20 129.2 43.2
21 139.2 53.2
22 150.2 53.6
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VIII. BIOGRAPHIES
Ahda P. Grilo (M’09) received her her Ph.D degree in Electrical
Engineering from the University of Campinas - UNICAMP, Brazil, in 2008.
Since 2009, she is an Assistant Professor at UFABC, Santo Andre, Brazil.
Currently she is a Post-Doctoral Fellow at the University of Alberta,
Edmonton, AB, Canada, on leave of absence from UFABC. Her interests
include analysis of distribution systems and distributed generation.
Wilsun Xu (M’90–SM’95–F’05) received the Ph.D. degree from the
University of British Columbia, Vancouver, Canada, in 1989. Currently, he is
a Professor and a NSERC/iCORE Industrial Research Chair at the University
of Alberta, Edmonton, Canada. His current research interests are power
quality, harmonics, and information extraction from power disturbances.
Madson C. de Almeida (M’07) received the M.Sc. and Ph.D. degrees in
electrical engineering from the State University of Campinas, Campinas,
Brazil, in 1999 and 2007, respectively. He is an Assistant Professor in the
Electrical Energy Systems Department at the State University of Campinas.
His research areas are power systems planning and control.